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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 726 · 40 questions
TalentJR

How to solve it (Think of the unluckiest case): Imagine the unluckiest draws: how many can you take and still NOT have what you want? Add 1 to that number — the next one must give it. Pigeonhole: with n boxes (colours, months, days), n × (k − 1) + 1 things make sure some box has at least k.

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  1. 1.A bag has 4 red balls and 3 white balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 4 (b) 2 (c) 7 (d) 5
  2. 2.A drawer has lots of black, green and red socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 6 (b) 3 (c) 2 (d) 4
  3. 3.A drawer has lots of blue, yellow, black and green socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 2 (b) 5 (c) 8 (d) 4
  4. 4.A bag has 12 green balls and 7 white balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 2 (b) 19 (c) 8 (d) 13
  5. 5.A drawer has lots of black, red, green and blue socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 2 (b) 4 (c) 5 (d) 8
  6. 6.A drawer has lots of yellow, white and green socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 6 (b) 3 (c) 2 (d) 4
  7. 7.A bag has 9 yellow balls and 8 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 2 (b) 10 (c) 11 (d) 17
  8. 8.What is the smallest number of children in a group that makes sure at least three of them were born in the same month? (a) 25 (b) 37 (c) 36 (d) 24
  9. 9.A drawer has lots of black, white, red, green and yellow socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 10 (b) 2 (c) 6 (d) 5
  10. 10.A drawer has lots of yellow and green socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 2 (b) 3 (c) 5 (d) 4
  11. 11.What is the smallest number of children in a group that makes sure at least three of them were born in the same day of the week? (a) 21 (b) 15 (c) 22 (d) 14
  12. 12.A drawer has lots of blue, white and black socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 4 (b) 2 (c) 3 (d) 6
  13. 13.A bag has 4 white balls and 5 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 2 (b) 9 (c) 5 (d) 6
  14. 14.A bag has 10 yellow balls and 8 black balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 18 (b) 11 (c) 9 (d) 2
  15. 15.A drawer has lots of black, green, blue and red socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 4 (b) 5 (c) 8 (d) 2
  16. 16.A bag has 8 white balls and 10 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 9 (b) 2 (c) 18 (d) 10
  17. 17.A bag has 7 white balls and 10 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 17 (b) 2 (c) 9 (d) 8
  18. 18.A drawer has lots of blue, white, red and yellow socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 2 (b) 8 (c) 4 (d) 5
  19. 19.A drawer has lots of red, yellow, black, white and blue socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 2 (b) 5 (c) 6 (d) 10
  20. 20.A bag has 4 red balls and 3 black balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 4 (b) 5 (c) 7 (d) 2
  21. 21.A bag has 3 red balls and 8 blue balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 11 (b) 2 (c) 9 (d) 4
  22. 22.A bag has 8 black balls and 9 red balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 2 (b) 17 (c) 10 (d) 9
  23. 23.A bag has 11 white balls and 6 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 12 (b) 2 (c) 13 (d) 17
  24. 24.A drawer has lots of black, white, red and yellow socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 2 (b) 4 (c) 8 (d) 5
  25. 25.A bag has 5 green balls and 7 black balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 2 (b) 6 (c) 12 (d) 8
  26. 26.A bag has 11 yellow balls and 8 green balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 9 (b) 19 (c) 2 (d) 12
  27. 27.A bag has 9 white balls and 10 black balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 2 (b) 10 (c) 19 (d) 11
  28. 28.A bag has 4 white balls and 8 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 2 (b) 12 (c) 5 (d) 6
  29. 29.A bag has 6 white balls and 7 black balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 8 (b) 7 (c) 2 (d) 13
  30. 30.A drawer has lots of green and white socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 4 (b) 5 (c) 3 (d) 2
  31. 31.A bag has 5 black balls and 10 green balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 6 (b) 2 (c) 15 (d) 11
  32. 32.What is the smallest number of children in a group that makes sure at least two of them were born in the same day of the week? (a) 14 (b) 8 (c) 15 (d) 7
  33. 33.What is the smallest number of children in a group that makes sure at least four of them were born in the same day of the week? (a) 21 (b) 22 (c) 28 (d) 29
  34. 34.A bag has 11 green balls and 4 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 15 (b) 13 (c) 12 (d) 2
  35. 35.What is the smallest number of children in a group that makes sure at least two of them were born in the same month? (a) 12 (b) 13 (c) 24 (d) 25
  36. 36.A bag has 4 blue balls and 3 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 2 (b) 6 (c) 7 (d) 5
  37. 37.A drawer has lots of green, white and blue socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 3 (b) 4 (c) 6 (d) 2
  38. 38.A bag has 10 green balls and 3 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 13 (b) 2 (c) 12 (d) 11
  39. 39.A drawer has lots of yellow, white, black and blue socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 2 (b) 5 (c) 8 (d) 4
  40. 40.A drawer has lots of blue and yellow socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 3 (b) 2 (c) 5 (d) 4

Answer key

  1. 5
  2. 4
  3. 5
  4. 13
  5. 5
  6. 4
  7. 11
  8. 25
  9. 6
  10. 3
  11. 15
  12. 4
  13. 6
  14. 11
  15. 5
  16. 10
  17. 9
  18. 5
  19. 6
  20. 5
  21. 9
  22. 10
  23. 13
  24. 5
  25. 8
  26. 12
  27. 11
  28. 6
  29. 8
  30. 3
  31. 11
  32. 8
  33. 22
  34. 13
  35. 13
  36. 6
  37. 4
  38. 12
  39. 5
  40. 3

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